Oooh, good one!
In Palo Alto, Ettan backs on a courtyard good for kids running about. I wouldn’t call it a playground.
With the “closed streets” of the peninsula, a wide number of restaurants could be said to have space next to tables without cars, but I think that’s cheating. You’re really asking about playgrounds.
JJ Cool Cafe at the Cantor art museum’s closer, because there’s a real outdoor space, but the restaurant is closed (and turned from JJ Cool to something else even before pandemic, if I remember right). There are a few other restaurants on campus, I want to say the Tall Tree, which are more realistically pro-kid, but campus is still closed and the restaurants may be as well.
The Mt View performing arts center has a cafe, the something bean, that fronts on a very large area that could almost be called a play area. I always liked the paninis but I wouldn’t call it great eats. You could certainly keep an eye on kids from the outdoor tables, but there are many others like that (the cafe in RWC on “theater way”). No play structures. Same with Cafe Borrone in MP, I have seen kids splashing around the fountain with parents eating at tables.
Pizza My Heart on University Ave in PA is right on Hamilton Park. It’s also a paved park, but at least boisterous.
Maybe we can count corporate cafes open to the public? I think the VMWare cafe is normally open to the public, and has a very nice feel and some playstructure-ish items — but also pandemic-closed.
I think my knowledge between 92 and 237 fairly encyclopedic, and I can’t think of any dining that is “really” on a playground.
With my niece and nephew, we’ve had playground meals, but it always has involved take-away from a nearby place, and a pick nick-style spread. One of the most memorable was Cuesta Park and Tommy’s Thai.
If you open the discussion to good-takeout-food-with-kid-friendly-items-paired-with-a-nearby-playground, I think I have entries.